Question $4$: $($recommended time: $40$min,$35pts)$

a$.$ Find the volume of the domain limited by the cylinder $x^2+y^2=1,$ the plane $y+z=2$ and $z0$

b$.$ Integrate $G(x,y,z)=x$ over the surface of the plane $y+z=2$ inside the cylinder $x^2+y^2=1.$

c$.$ Use the surface integral in Stoke's theorem to calculate the circulation of the vector field vec$(F)$ around the curve $(C)$ where vec$(F)=-y^2$vec$()+$xvec$(j)+z^2$vec$(k).($C$)$ the intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1,$ counterclockwise when viewed from above.

Answer

a$.$ Find the volume of the domain limited by the cylinder $x^2+y^2=1,$ the plane $y+z=2$ and $z0$

b$.$ Integrate $G(x,y,z)=x$ over the surface of the plane $y+z=2$ inside the cylinder $x^2+y^2=1.$

c$.$ Use the surface integral in Stoke's theorem to calculate the circulation of the vector field around the curve $(C)$ where vec$(F)=-y^2$vec$()+$xvec$(j)+z^2$vec$(k).($C$)$ the intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1,$ counterclockwise when viewed from above.